The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization
This addresses the convergence behavior of first-order methods in min-max problems, relevant for applications like GAN training, but is incremental in extending dynamical systems analysis.
The paper characterizes the limit points of Gradient Descent/Ascent (GDA) and Optimistic Gradient Descent Ascent (OGDA) in min-max optimization, showing that both methods avoid unstable critical points for almost all initializations and that OGDA-stable points are a superset of GDA-stable points, which include local min-max solutions.
Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that they may cycle, and there is no good understanding of their limit points when they do not. When they converge, do they converge to local min-max solutions? We characterize the limit points of two basic first order methods, namely Gradient Descent/Ascent (GDA) and Optimistic Gradient Descent Ascent (OGDA). We show that both dynamics avoid unstable critical points for almost all initializations. Moreover, for small step sizes and under mild assumptions, the set of \{OGDA\}-stable critical points is a superset of \{GDA\}-stable critical points, which is a superset of local min-max solutions (strict in some cases). The connecting thread is that the behavior of these dynamics can be studied from a dynamical systems perspective.